Transformation Exercises: Denavit-

Hartenberg Method

Some images and exercises from: Introduction to Autonomous Mobile Robots, Siegwart, Nourbakhsh, 2011

Robot Dynamics and Control Second Edition, Spong, Hutchinson, Vidyasagar, 2004Spacecraft Robot Kinematics Using Dual Quaternions, Valverde, Alfredo & Tsiotras, Panagiotis, 2018

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§ Given two frames of reference, transformationsconvert configurations (position + orientation) from one to the other.§ A robot sees a thing. Where is the thing in the world?§ There’s a thing in the world. Where is it wrt. the robot?§ A robot moves around. Where is it in the world?

§ We can do this with translation/rotation matrices, multiplied by configuration.§ 𝜉" = $

"𝑇 × 𝜉$§ Or, faster with the Denavit-Hartenberg method.

Review

https://matrix.reshish.com/multiplication.php is nice for running matrix multiplications

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§ z axis is axis of motion§ Rotation around z for revolute§ Translation along z for prismatic

§ xi axis orthogonal to zi and zi-1

§ y axis: right-hand rule§ Fingers point +x§ Thumb points +z§ Palm faces +y

§ xi axis must intersect zi-1 axis § Which may mean translating origin

DH review: frames +x

+z

+y

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§ ai-1 : link lengthdistance Zi-1 ⇆ Zialong Xi

§ αi-1 : link twistangle between Zi-1⇆ Zi around Xi

§ di : link offsetdistance Xi-1 to Xialong Zi

§ θi : joint angle angle between Xi-1and Xi around Zi

DH review: parameters

Valverde, Alfredo & Tsiotras, Panagiotis. (2018). Spacecraft Robot Kinematics Using Dual Quaternions.https://www.mdpi.com/2218-6581/7/4/64/htm

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Review: Transformation matrices

Rii−1 =

cosθi −sinθi cosαi,i+1 sinθi sinαi,i+1 ai,i+1 cosθisinθi cosθi cosαi,i+1 −cosθi sinαi,i+1 ai,i+1 sinθi0 sinαi,i+1 cosαi,i+1 di0 0 0 1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

xyz

Tx

xyzTy

xyzTz

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§ Define the axes according to the DH rules.

§ Then, draw lines and arcs for a and θ.§ Why not α and d?

Ex.1: Planar elbow manipulator

How many frames of reference do you need?

One per joint, so 2.

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Exercise 1

Z axes point out towards us

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§ Give the DH parameters.

Exercise 1

Z axes point out towards us

First, what’s the table?

What are the values?

Planar arm makes params simpler, so…

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§ Give the final transformation matrix.

Exercise 1

Rii−1 =

cosθi −sinθi cosαi,i+1 sinθi sinαi,i+1 ai,i+1 cosθisinθi cosθi cosαi,i+1 −cosθi sinαi,i+1 ai,i+1 sinθi0 sinαi,i+1 cosαi,i+1 di0 0 0 1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

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§ Give the final transformation matrix.

§ Plugging in…

Exercise 1

Rii−1 =

cosθi −sinθi cosαi,i+1 sinθi sinαi,i+1 ai,i+1 cosθisinθi cosθi cosαi,i+1 −cosθi sinαi,i+1 ai,i+1 sinθi0 sinαi,i+1 cosαi,i+1 di0 0 0 1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

𝑇() =

cos 𝜃( −sin 𝜃( 0 𝑎),(cos 𝜃(sin 𝜃( cos 𝜃( 0 𝑎),(sin 𝜃(0 0 1 00 0 0 1

𝑇5( =

cos 𝜃5 −sin 𝜃5 0 𝑎(,5cos 𝜃5sin 𝜃5 cos 𝜃5 0 𝑎(,5 sin 𝜃50 0 1 00 0 0 1

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§ Give the final transformation matrix.

§ Plugging in…

Exercise 1

Rii−1 =

cosθi −sinθi cosαi,i+1 sinθi sinαi,i+1 ai,i+1 cosθisinθi cosθi cosαi,i+1 −cosθi sinαi,i+1 ai,i+1 sinθi0 sinαi,i+1 cosαi,i+1 di0 0 0 1

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

𝑇5) = 𝑇()×𝑇5(=

cos(𝜃( + 𝜃5) −sin(𝜃( + 𝜃5) 0 𝑎),(cos 𝜃( +𝑎(,5cos(𝜃( + 𝜃5)sin(𝜃( + 𝜃5) cos(𝜃( + 𝜃5) 0 𝑎),(sin 𝜃( +𝑎(,5sin(𝜃( + 𝜃5)

0 0 1 00 0 0 1

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§ This is always the parameters and transform for a planar elbow manipulator.

§ Similar derivations can be done for cylindrical arms, spherical arms, etc.§ This is why we name configurations.

§ If you know these, you can subdivide an arm.

Exercise 1

cos(𝜃( + 𝜃5) −sin(𝜃( + 𝜃5) 0 𝑎),(cos 𝜃( +𝑎(,5cos(𝜃( + 𝜃5)sin(𝜃( + 𝜃5) cos(𝜃( + 𝜃5) 0 𝑎),(sin 𝜃( +𝑎(,5sin(𝜃( + 𝜃5)

0 0 1 00 0 0 1

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§ Could you do the same thing using a sequence of x/y/z rotations and translations?§ How many steps would it take?

§ Could you do the same thing given a word problem about an arm in the world?§ Given numbers?§ As a derivation?

§ How many joints would you max out at?§ Directly?§ Using DH parameters?

Discussion

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1. What is the complete, derived transformation matrix for a spherical wrist?

a. What are the frames?b. What are the

DH parameters?c. What are the

individual transformation matrices?d. What’s the final transformation matrix?

2. What is the final transformation matrix for this wrist, without using the DHmethod? (It’s the same)

§ This is just a sequence of rotations – see Spong.

Exercise 2:

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§ What is the complete, derived transformation matrix for a 3-link cylindrical robot?a. What are the frames?b. What are the DH parameters?c. What are the individual

transformation matrices?d. What’s the final transformation

matrix?e. What is the final transformation

matrix for this wrist, derived withoutusing the DH method?

§ See Spong.

Exercise 3:

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a. What are the frames?

Exercise 3:

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a. What are the frames?

§ Joint 0:§ z0 is along axis of motion§ Origin 0 (O0) is arbitrary,

but makes sense§ x0 is normal to the page.

Exercise 3:

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a. What are the frames?

§ Joint 1:§ z1 is along axis of motion§ Origin 0 (O1) is easy,

because z0 and z1 are the same (no origin movement necessary)

§ x1 Is normal to the page currently (but not when joint 0 moves!)

Exercise 3:

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a. What are the frames?

§ Joint 2:§ z2 is along axis of motion§ x2 is chosen parallel to x1

so that θ2 is zero.

§ Joint 3: § Chosen as shown.

Exercise 3:

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b. What are the DH parameters?

Exercise 3:

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c. What are theindividual transformation matrices?

§ c = cosine, s = sin

Exercise 3:

0T1 =

1T2 =

2T3 =

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d. What’s the final transformation matrix?

Exercise 3:

0T3 =

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§ What is the complete, derived transformation matrix for this arm? (any approach)

Exercise 4:

§ This is exercise 3 and exercise 2 stuck together, so you can just multiply your transforms for those!